Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The average Mordell-Weil rank of elliptic surfaces over number fields

Remke Kloosterman

TL;DR

The paper proves that for families of elliptic surfaces over finitely generated fields with fixed singular-fiber configuration, the locus where the Mordell-Weil rank jumps is sparse, implying the average rank is zero (Cowan's conjecture) over $\mathbf{Q}$ and extends to number fields. The core strategy combines specialization maps, Maulik–Poonen results on specialization, and a quadratic-twisting argument to force finiteness of the generic Mordell-Weil group on a dense set. It also constructs universal Weierstrass models to control the trivial lattice and uses thin-set density results to transfer local finiteness to global averages. The paper further illustrates that, while the average rank is zero in general, there exist explicit families with positive average rank, highlighting the role of twists and base-field arithmetic. Overall, it extends Cowan's conjecture to arbitrary number fields and clarifies when zero-average behavior holds in elliptic-surface families.

Abstract

Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside $|\mathcal{B}|$ where the Mordell-Weil rank is at least $r+1$ is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over $\mathbb{Q}$ and prove a similar result for elliptic surfaces over arbitrary number fields.

The average Mordell-Weil rank of elliptic surfaces over number fields

TL;DR

The paper proves that for families of elliptic surfaces over finitely generated fields with fixed singular-fiber configuration, the locus where the Mordell-Weil rank jumps is sparse, implying the average rank is zero (Cowan's conjecture) over and extends to number fields. The core strategy combines specialization maps, Maulik–Poonen results on specialization, and a quadratic-twisting argument to force finiteness of the generic Mordell-Weil group on a dense set. It also constructs universal Weierstrass models to control the trivial lattice and uses thin-set density results to transfer local finiteness to global averages. The paper further illustrates that, while the average rank is zero in general, there exist explicit families with positive average rank, highlighting the role of twists and base-field arithmetic. Overall, it extends Cowan's conjecture to arbitrary number fields and clarifies when zero-average behavior holds in elliptic-surface families.

Abstract

Let be a finitely generated field over . Let be a family of elliptic surfaces over such that each elliptic fibration has the same configuration of singular fibers. Let be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside where the Mordell-Weil rank is at least is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over and prove a similar result for elliptic surfaces over arbitrary number fields.
Paper Structure (3 sections, 21 theorems, 37 equations)

This paper contains 3 sections, 21 theorems, 37 equations.

Table of Contents

  1. introduction
  2. Families of elliptic surfaces
  3. Cowan's conjecture

Key Result

Theorem 1.2

Let $K$ be a finitely generated field over $\mathbf{Q}$, and let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces with a section over $K$. Then there exists a sparse set $Z\subset |\mathcal{B}|$ such that for all $b\in |\mathcal{B}|\setminus Z$ the specialisation maps and are bijective.

Theorems & Definitions (51)

  • Conjecture 1.1: Cowan
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 41 more